\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  11.0 & -3.00 x_{1} & +  7.00 x_{2} & -7.00 x_{3} & +  8.00 x_{4} & +  2.00 x_{5} & +  2.00 x_{6}\\
 x_{8}   &  -11.0 & +  9.00 x_{1} & +  8.00 x_{2} & -6.00 x_{3} & -8.00 x_{4} & -9.00 x_{5} & +  4.00 x_{6}\\
 x_{9}   &  -6.0 & +  1.00 x_{1} & +  9.00 x_{2} & -6.00 x_{3} & -10.00 x_{4} & +  9.00 x_{5} & + 10.00 x_{6}\\
 x_{10}   &  6.0 & -4.00 x_{1} & -6.00 x_{2} & +  3.00 x_{3} & +  5.00 x_{4} & + 10.00 x_{5} & -5.00 x_{6}\\
 x_{11}   &  13.0 & -3.00 x_{1} & +  6.00 x_{2} & -10.00 x_{3} & +  8.00 x_{4} & +  1.00 x_{5} & +  1.00 x_{6}\\
 x_{12}   &  32.0 & -6.00 x_{1} & -8.00 x_{2} & -6.00 x_{3} & -9.00 x_{4} &   & -3.00 x_{6}\\
 x_{13}   &  -13.0 & +  5.00 x_{1} & +  5.00 x_{2} & +  6.00 x_{3} & +  7.00 x_{4} & -7.00 x_{5} & -4.00 x_{6}\\
\hline
z    &  0.0 & -3.00 x_{1} & -3.00 x_{2} & -5.00 x_{3} & -1.00 x_{4} & +  5.00 x_{5} & -5.00 x_{6}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{6}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,13$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & +  3.00 y_{7} & -9.00 y_{8} & -1.00 y_{9} & +  4.00 y_{10} & +  3.00 y_{11} & +  6.00 y_{12} & -5.00 y_{13}\\
 y_{2}   &  1.0 & -7.00 y_{7} & -8.00 y_{8} & -9.00 y_{9} & +  6.00 y_{10} & -6.00 y_{11} & +  8.00 y_{12} & -5.00 y_{13}\\
 y_{3}   &  1.0 & +  7.00 y_{7} & +  6.00 y_{8} & +  6.00 y_{9} & -3.00 y_{10} & + 10.00 y_{11} & +  6.00 y_{12} & -6.00 y_{13}\\
 y_{4}   &  1.0 & -8.00 y_{7} & +  8.00 y_{8} & + 10.00 y_{9} & -5.00 y_{10} & -8.00 y_{11} & +  9.00 y_{12} & -7.00 y_{13}\\
 y_{5}   &  1.0 & -2.00 y_{7} & +  9.00 y_{8} & -9.00 y_{9} & -10.00 y_{10} & -1.00 y_{11} &   & +  7.00 y_{13}\\
 y_{6}   &  1.0 & -2.00 y_{7} & -4.00 y_{8} & -10.00 y_{9} & +  5.00 y_{10} & -1.00 y_{11} & +  3.00 y_{12} & +  4.00 y_{13}\\
\hline
z    &  -0 & -11.00 y_{7} & + 11.00 y_{8} & +  6.00 y_{9} & -6.00 y_{10} & -13.00 y_{11} & -32.00 y_{12} & + 13.00 y_{13}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 5 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{8}   &  0.111371324486 & +  1.58 y_{7} & -0.21 y_{1} & +  0.10 y_{2} & -0.04 y_{5} & +  1.51 y_{11} & +  0.11 y_{12} & +  0.04 y_{4}\\
 y_{9}   &  0.08014571949 & -0.79 y_{7} & +  0.04 y_{1} & -0.08 y_{2} & -0.03 y_{5} & -0.67 y_{11} & +  0.45 y_{12} & -0.00 y_{4}\\
 y_{3}   &  0.181889149102 & + 15.90 y_{7} & -0.22 y_{1} & +  0.27 y_{2} & -0.03 y_{5} & + 18.65 y_{11} & -2.00 y_{12} & +  0.80 y_{4}\\
 y_{13}   &  0.195420244601 & -1.23 y_{7} & -0.02 y_{1} & -0.08 y_{2} & -0.01 y_{5} & -1.13 y_{11} & +  1.52 y_{12} & -0.09 y_{4}\\
 y_{10}   &  0.264897215717 & +  1.07 y_{7} & -0.24 y_{1} & +  0.11 y_{2} & -0.11 y_{5} & +  1.07 y_{11} & +  0.75 y_{12} & -0.03 y_{4}\\
 y_{6}   &  1.85922456414 & -0.01 y_{7} & -0.80 y_{1} & +  0.66 y_{2} & -0.12 y_{5} & +  0.45 y_{11} & +  7.88 y_{12} & -0.60 y_{4}\\
\hline
z    &  2.65703877179 & -20.72 y_{7} & -0.89 y_{1} & -1.08 y_{2} & -0.12 y_{5} & -21.52 y_{11} & -12.89 y_{12} & -0.57 y_{4}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  20.7243039292 & -1.58 x_{8} & +  0.79 x_{9} & -15.90 x_{3} & +  1.23 x_{13} & -1.07 x_{10} & +  0.01 x_{6}\\
 x_{1}   &  0.885115794952 & +  0.21 x_{8} & -0.04 x_{9} & +  0.22 x_{3} & +  0.02 x_{13} & +  0.24 x_{10} & +  0.80 x_{6}\\
 x_{2}   &  1.08300806661 & -0.10 x_{8} & +  0.08 x_{9} & -0.27 x_{3} & +  0.08 x_{13} & -0.11 x_{10} & -0.66 x_{6}\\
 x_{5}   &  0.118787405673 & +  0.04 x_{8} & +  0.03 x_{9} & +  0.03 x_{3} & +  0.01 x_{13} & +  0.11 x_{10} & +  0.12 x_{6}\\
 x_{11}   &  21.5225084569 & -1.51 x_{8} & +  0.67 x_{9} & -18.65 x_{3} & +  1.13 x_{13} & -1.07 x_{10} & -0.45 x_{6}\\
 x_{12}   &  12.8940931564 & -0.11 x_{8} & -0.45 x_{9} & +  2.00 x_{3} & -1.52 x_{13} & -0.75 x_{10} & -7.88 x_{6}\\
 x_{4}   &  0.570127504554 & -0.04 x_{8} & +  0.00 x_{9} & -0.80 x_{3} & +  0.09 x_{13} & +  0.03 x_{10} & +  0.60 x_{6}\\
\hline
z    &  -2.65703877179 & -0.11 x_{8} & -0.08 x_{9} & -0.18 x_{3} & -0.20 x_{13} & -0.26 x_{10} & -1.86 x_{6}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  20.7243039292 & -1.58 x_{8} & +  0.79 x_{9} & -15.90 x_{3} & +  1.23 x_{13} & -1.07 x_{10} & +  0.01 x_{6}\\
 x_{1}   &  0.885115794952 & +  0.21 x_{8} & -0.04 x_{9} & +  0.22 x_{3} & +  0.02 x_{13} & +  0.24 x_{10} & +  0.80 x_{6}\\
 x_{2}   &  1.08300806661 & -0.10 x_{8} & +  0.08 x_{9} & -0.27 x_{3} & +  0.08 x_{13} & -0.11 x_{10} & -0.66 x_{6}\\
 x_{5}   &  0.118787405673 & +  0.04 x_{8} & +  0.03 x_{9} & +  0.03 x_{3} & +  0.01 x_{13} & +  0.11 x_{10} & +  0.12 x_{6}\\
 x_{11}   &  21.5225084569 & -1.51 x_{8} & +  0.67 x_{9} & -18.65 x_{3} & +  1.13 x_{13} & -1.07 x_{10} & -0.45 x_{6}\\
 x_{12}   &  12.8940931564 & -0.11 x_{8} & -0.45 x_{9} & +  2.00 x_{3} & -1.52 x_{13} & -0.75 x_{10} & -7.88 x_{6}\\
 x_{4}   &  0.570127504554 & -0.04 x_{8} & +  0.00 x_{9} & -0.80 x_{3} & +  0.09 x_{13} & +  0.03 x_{10} & +  0.60 x_{6}\\
\hline
z    &  -5.88056206089 & -0.08 x_{8} & +  0.03 x_{9} & -3.93 x_{3} & -0.32 x_{13} & +  0.17 x_{10} & -5.39 x_{6}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  20.7243039292 & -1.58 x_{8} & +  0.79 x_{9} & -15.90 x_{3} & +  1.23 x_{13} & -1.07 x_{10} & +  0.01 x_{6}\\
 x_{1}   &  0.885115794952 & +  0.21 x_{8} & -0.04 x_{9} & +  0.22 x_{3} & +  0.02 x_{13} & +  0.24 x_{10} & +  0.80 x_{6}\\
 x_{2}   &  1.08300806661 & -0.10 x_{8} & +  0.08 x_{9} & -0.27 x_{3} & +  0.08 x_{13} & -0.11 x_{10} & -0.66 x_{6}\\
 x_{5}   &  0.118787405673 & +  0.04 x_{8} & +  0.03 x_{9} & +  0.03 x_{3} & +  0.01 x_{13} & +  0.11 x_{10} & +  0.12 x_{6}\\
 x_{11}   &  21.5225084569 & -1.51 x_{8} & +  0.67 x_{9} & -18.65 x_{3} & +  1.13 x_{13} & -1.07 x_{10} & -0.45 x_{6}\\
 x_{12}   &  12.8940931564 & -0.11 x_{8} & -0.45 x_{9} & +  2.00 x_{3} & -1.52 x_{13} & -0.75 x_{10} & -7.88 x_{6}\\
 x_{4}   &  0.570127504554 & -0.04 x_{8} & +  0.00 x_{9} & -0.80 x_{3} & +  0.09 x_{13} & +  0.03 x_{10} & +  0.60 x_{6}\\
\hline
z    &  -5.88056206089 & -0.08 x_{8} & +  0.03 x_{9} & -3.93 x_{3} & -0.32 x_{13} & +  0.17 x_{10} & -5.39 x_{6}\\
\end{array}\]


 $ x_{9} $ enters and $ x_{1} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  38.488372093 & +  2.67 x_{8} & -20.07 x_{1} & -11.40 x_{3} & +  1.56 x_{13} & +  3.70 x_{10} & + 16.02 x_{6}\\
 x_{9}   &  22.6013289037 & +  5.41 x_{8} & -25.53 x_{1} & +  5.73 x_{3} & +  0.42 x_{13} & +  6.06 x_{10} & + 20.37 x_{6}\\
 x_{2}   &  2.97674418605 & +  0.35 x_{8} & -2.14 x_{1} & +  0.21 x_{3} & +  0.12 x_{13} & +  0.40 x_{10} & +  1.05 x_{6}\\
 x_{5}   &  0.880398671096 & +  0.22 x_{8} & -0.86 x_{1} & +  0.22 x_{3} & +  0.03 x_{13} & +  0.32 x_{10} & +  0.81 x_{6}\\
 x_{11}   &  36.6312292359 & +  2.10 x_{8} & -17.07 x_{1} & -14.82 x_{3} & +  1.42 x_{13} & +  2.98 x_{10} & + 13.17 x_{6}\\
 x_{12}   &  2.68438538206 & -2.55 x_{8} & + 11.53 x_{1} & -0.59 x_{3} & -1.71 x_{13} & -3.49 x_{10} & -17.08 x_{6}\\
 x_{4}   &  0.611295681063 & -0.03 x_{8} & -0.05 x_{1} & -0.79 x_{3} & +  0.09 x_{13} & +  0.04 x_{10} & +  0.63 x_{6}\\
\hline
z    &  -5.13953488372 & +  0.09 x_{8} & -0.84 x_{1} & -3.74 x_{3} & -0.30 x_{13} & +  0.37 x_{10} & -4.72 x_{6}\\
\end{array}\]


 $ x_{8} $ enters and $ x_{12} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  41.3020833333 & -1.05 x_{12} & -7.98 x_{1} & -12.01 x_{3} & -0.23 x_{13} & +  0.04 x_{10} & -1.88 x_{6}\\
 x_{9}   &  28.2916666667 & -2.12 x_{12} & -1.08 x_{1} & +  4.48 x_{3} & -3.20 x_{13} & -1.34 x_{10} & -15.84 x_{6}\\
 x_{2}   &  3.34375 & -0.14 x_{12} & -0.56 x_{1} & +  0.13 x_{3} & -0.12 x_{13} & -0.08 x_{10} & -1.29 x_{6}\\
 x_{5}   &  1.11458333333 & -0.09 x_{12} & +  0.15 x_{1} & +  0.17 x_{3} & -0.12 x_{13} & +  0.01 x_{10} & -0.68 x_{6}\\
 x_{11}   &  38.84375 & -0.82 x_{12} & -7.56 x_{1} & -15.31 x_{3} & +  0.01 x_{13} & +  0.11 x_{10} & -0.91 x_{6}\\
 x_{8}   &  1.05208333333 & -0.39 x_{12} & +  4.52 x_{1} & -0.23 x_{3} & -0.67 x_{13} & -1.37 x_{10} & -6.70 x_{6}\\
 x_{4}   &  0.583333333333 & +  0.01 x_{12} & -0.17 x_{1} & -0.78 x_{3} & +  0.10 x_{13} & +  0.07 x_{10} & +  0.81 x_{6}\\
\hline
z    &  -5.04166666667 & -0.04 x_{12} & -0.42 x_{1} & -3.77 x_{3} & -0.36 x_{13} & +  0.24 x_{10} & -5.34 x_{6}\\
\end{array}\]


 $ x_{10} $ enters and $ x_{8} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  41.3311132255 & -1.06 x_{12} & -7.85 x_{1} & -12.02 x_{3} & -0.25 x_{13} & -0.03 x_{8} & -2.07 x_{6}\\
 x_{9}   &  27.2626070409 & -1.74 x_{12} & -5.51 x_{1} & +  4.71 x_{3} & -2.54 x_{13} & +  0.98 x_{8} & -9.29 x_{6}\\
 x_{2}   &  3.28068506185 & -0.11 x_{12} & -0.83 x_{1} & +  0.14 x_{3} & -0.08 x_{13} & +  0.06 x_{8} & -0.89 x_{6}\\
 x_{5}   &  1.12559467174 & -0.09 x_{12} & +  0.19 x_{1} & +  0.17 x_{3} & -0.13 x_{13} & -0.01 x_{8} & -0.75 x_{6}\\
 x_{11}   &  38.9248334919 & -0.85 x_{12} & -7.21 x_{1} & -15.33 x_{3} & -0.04 x_{13} & -0.08 x_{8} & -1.43 x_{6}\\
 x_{10}   &  0.768791627022 & -0.29 x_{12} & +  3.30 x_{1} & -0.17 x_{3} & -0.49 x_{13} & -0.73 x_{8} & -4.89 x_{6}\\
 x_{4}   &  0.639391056137 & -0.01 x_{12} & +  0.07 x_{1} & -0.79 x_{3} & +  0.07 x_{13} & -0.05 x_{8} & +  0.46 x_{6}\\
\hline
z    &  -4.85347288297 & -0.11 x_{12} & +  0.39 x_{1} & -3.81 x_{3} & -0.48 x_{13} & -0.18 x_{8} & -6.54 x_{6}\\
\end{array}\]


 $ x_{1} $ enters and $ x_{2} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  10.4155251142 & +  0.01 x_{12} & +  9.42 x_{2} & -13.36 x_{3} & +  0.48 x_{13} & -0.59 x_{8} & +  6.30 x_{6}\\
 x_{9}   &  5.59360730594 & -0.99 x_{12} & +  6.61 x_{2} & +  3.77 x_{3} & -2.03 x_{13} & +  0.58 x_{8} & -3.43 x_{6}\\
 x_{1}   &  3.93607305936 & -0.14 x_{12} & -1.20 x_{2} & +  0.17 x_{3} & -0.09 x_{13} & +  0.07 x_{8} & -1.07 x_{6}\\
 x_{5}   &  1.88584474886 & -0.12 x_{12} & -0.23 x_{2} & +  0.20 x_{3} & -0.15 x_{13} & +  0.00 x_{8} & -0.96 x_{6}\\
 x_{11}   &  10.5296803653 & +  0.13 x_{12} & +  8.66 x_{2} & -16.56 x_{3} & +  0.62 x_{13} & -0.60 x_{8} & +  6.25 x_{6}\\
 x_{10}   &  13.7716894977 & -0.74 x_{12} & -3.96 x_{2} & +  0.40 x_{3} & -0.79 x_{13} & -0.49 x_{8} & -8.41 x_{6}\\
 x_{4}   &  0.931506849315 & -0.02 x_{12} & -0.09 x_{2} & -0.78 x_{3} & +  0.06 x_{13} & -0.05 x_{8} & +  0.38 x_{6}\\
\hline
z    &  -3.31050228311 & -0.16 x_{12} & -0.47 x_{2} & -3.74 x_{3} & -0.52 x_{13} & -0.15 x_{8} & -6.96 x_{6}\\
\end{array}\]
Final Dictionary
Solution:  -3.31050228311
Num Pivots:  5
\end{document}
